# Stability and bifurcation analysis on a Logistic model with discrete and distributed delays.

**ABSTRACT** In this paper, a Logistic model with discrete and distributed delays is investigated. We first consider the stability of the positive equilibrium and the existence of local Hopf bifurcations. In succession, using the normal form theory and center manifold argument, we derive the explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions. A numerical simulation for supporting the theoretical analysis is also given.

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**ABSTRACT:**The problem of BIBO stabilization for multiple mixed time-delayed control system with nonlinear perturbations is studied in this paper. The new delay-dependent BIBO stabilization criteria are derived by the Lyapunov functional and given in terms of existence of a positive definite solution to an auxiliary algebraic Riccati matrix equation. The robust quadratic stability for such system is also discussed. The work of this paper will extend the results of some references.Applied Mathematics and Computation. 01/2008; 196:207-213. - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, we investigate the stability and Hopf bifurcation of a new regulated logistic growth with discrete and distributed delays. By choosing the discrete delay τ as a bifurcation parameter, we prove that the system is locally asymptotically stable in a range of the delay and Hopf bifurcation occurs as τ crosses a critical value. Furthermore, explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Finally, an illustrative example is also given to support the theoretical results.Communications in Nonlinear Science and Numerical Simulation 01/2009; 14(12):4292-4303. · 2.77 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, the dynamical behavior of an eco-epidemiological model with discrete and distributed delay is studied. Sufficient conditions for the local asymptotical stability of the nonnegative equilibria are obtained. We prove that there exists a threshold value of the feedback time delay τ beyond which the positive equilibrium bifurcates towards a periodic solution. Using the normal form theory and center manifold argument, the explicit formulae which determine the stability, the direction and the periodic of bifurcating period solutions are derived. Numerical simulations are carried out to explain the mathematical conclusions.Journal of Applied Mathematics and Computing 01/2010; 33(1):305-325.

Page 1

Stability and bifurcation analysis on a Logistic model

with discrete and distributed delaysq

Yongli Songa,*, Yahong Pengb

aDepartment of Applied Mathematics, Tongji University, 1239 Siping Road, Shanghai 200092, PR China

bDepartment of Applied Mathematics, Donghua University, Shanghai 200051, PR China

Abstract

In this paper, a Logistic model with discrete and distributed delays is investigated. We first consider the stability of the

positive equilibrium and the existence of local Hopf bifurcations. In succession, using the normal form theory and center

manifold argument, we derive the explicit formulas which determine the stability, direction and other properties of bifur-

cating periodic solutions. A numerical simulation for supporting the theoretical analysis is also given.

? 2006 Elsevier Inc. All rights reserved.

Keywords: Logistic model; Time delays; Stability; Hopf bifurcations; Periodic solutions

1. Introduction

The well-known Logistic differential equation with discrete delay

?

and the distributed delay Logistic equation

Zt

have been widely studied (see for example [1–6] and references therein). In addition, Gopalsamy [7] also stud-

ied a Logistic model with two delays

_ xðtÞ ¼ rxðtÞ 1 ?xðt ? sÞ

K

?

ð1:1Þ

_ xðtÞ ¼ rxðtÞ 1 ?xðtÞ

K

?1

Q

?1

fðt ? sÞxðsÞ ds

??

ð1:2Þ

_ xðtÞ ¼ rxðtÞð1 ? a1xðt ? sÞ ? a2xðt ? cÞÞ:

In the present paper, we consider the following more general delayed Logistic equation

ð1:3Þ

0096-3003/$ - see front matter ? 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2006.03.025

qThis research is supported in part by the National Natural Science Foundation of China.

*Corresponding author.

E-mail address: syl.mail@163.com (Y. Song).

Applied Mathematics and Computation 181 (2006) 1745–1757

www.elsevier.com/locate/amc

Page 2

_ xðtÞ ¼ rxðtÞ 1 ? a1xðt ? sÞ ? a2

Zt

?1

fðt ? sÞxðsÞ ds

??

;

ð1:4Þ

where r,s,a1,a2> 0. The function f in (1.4) is called the delayed kernel, which is the weight given to the pop-

ulation t time units ago, and we shall assume it satisfies f(t) P 0 for all t P 0 together with the normalization

condition

Z1

which ensures that the steady states of the model (1.4) are unaffected by the delay. Clearly, systems (1.1)–(1.3)

are the special cases of system (1.4) under some special delayed kernel f(t) such as when f(t) = d(t ? r), Eq.

(1.4) becomes Eq. (1.3). Following the ideal of Cushing [8], the weak kernel

fðtÞ ¼ re?rt;

and the strong kernel

0

fðtÞ dt ¼ 1;

r > 0

ð1:5Þ

fðtÞ ¼ r2te?rt;

are frequently encountered in the literature. In this paper, although we consider only Eq. (1.4) with the weak

kernel (1.5), the strong kernel case can be handled similarly. Taking the delay s as a parameter, we investigate

the effect of the delay s on the dynamics of Eq. (1.4). More specifically, we show that the stability switches and

a Hopf bifurcation occur when the delay s passes through a critical value.

This paper is organized as follows. In Section 2, we consider the stability and the local Hopf bifurcation of

the positive equilibrium. In Section 3, we first employ the normal form method and the center manifold theory

introduced by Hassard et al. [9] to analyze the direction, stability and the period of the bifurcating periodic

solution at the critical values of s, then a numerical example is given. We end with some conlusions in Section

4.

r > 0

ð1:6Þ

2. Stability of the positive equilibrium and existence of Hopf bifurcations

Clearly, Eq. (1.4) always has a positive equilibrium

x?¼

1

a1þ a2:

Now, we define a new variable

Zt

then by the linear chain trick technique, system (1.4) can be transformed into the following equivalent system

with a discrete delay

?

Obviously, system (2.1) has a unique positive equilibrium E*= (x*,y*) with y*= x*. By the linear transform

?

system (2.1) becomes

?

Then the linearized system of (2.2) at the zero steady state is

yðtÞ ¼

?1

re?rðt?sÞxðsÞ ds;

_ xðtÞ ¼ rxðtÞð1 ? a1xðt ? sÞ ? a2yðtÞÞ;

_ yðtÞ ¼ rxðtÞ ? ryðtÞ:

ð2:1Þ

x1ðtÞ ¼ xðtÞ ? x?;

x2ðtÞ ¼ yðtÞ ? y?;

_ x1ðtÞ ¼ ?ra1x?x1ðt ? sÞ ? ra2x?x2ðtÞ ? ra1x1ðtÞx1ðt ? sÞ ? ra2x1ðtÞx2ðtÞ;

_ x2ðtÞ ¼ rx1ðtÞ ? rx2ðtÞ:

ð2:2Þ

1746

Y. Song, Y. Peng / Applied Mathematics and Computation 181 (2006) 1745–1757

Page 3

_ x1ðtÞ ¼ ?ra1x?x1ðt ? sÞ ? ra2x?x2ðtÞ;

_ x2ðtÞ ¼ rx1ðtÞ ? rx2ðtÞ:

The associated characteristic equation of system (2.3) is the following second degree exponential polynomial

equation:

?

ð2:3Þ

k2þ pk þ g þ ðsk þ qÞe?ks¼ 0;

where p = r > 0, g = rra2x*> 0, s = ra1x*> 0, q = rra1x*> 0.

In order to investigate the stability of the positive equilibrium of system (2.1), we need to study the distri-

bution of roots of Eq. (2.4). Clearly, k = 0 is not a root of Eq. (2.4).

If ix(x > 0) is a root of Eq. (2.4), then

ð2:4Þ

g ? x2þ sxsinxs þ qcosxs þ pxi þ ½sxcosxs ? qsinxs?i ¼ 0:

Separating the real and imaginary parts, we have

?

which lead to

x2? g ¼ sxsinxs þ qcosxs;

px ¼ qsinxs ? sxcosxs;

ð2:5Þ

x4? ðs2þ 2g ? p2Þx2þ g2? q2¼ 0:

ð2:6Þ

Let

D ¼ ðs2þ 2g ? p2Þ2? 4ðg2? q2Þ:

It is easy to see that if at least one of the following is satisfied:

ðH1Þ

ðH2Þ

ðH3Þ

D < 0;

D > 0;

g2? q2P 0;

s2þ 2g ? p26 0;

s2þ 2g ? p2< 0;

D ¼ 0;

then Eq. (2.6) has no positive root; if

D > 0;

g2? q2> 0;

s2þ 2g ? p2> 0

or

D ¼ 0;

s2þ 2g ? p2> 0

holds, then Eq. (2.6) has two positive roots

ffiffiffi

and if

x?¼

2

p

2

s2þ 2g ? p2?

ffiffiffiffi

D

p

hi1

2;

g2? q2< 0

or

g2? q2¼ 0;

holds, then Eq. (2.6) has only one positive root x+.

Without loss of generality, we suppose that Eq. (2.6) has two positive roots ±x±. Then, from (2.5), we can

determine

ðq ? psÞx2

s2x2

s2þ 2g ? p2> 0

s?

j¼

1

x?

cos?1

?? gq

?þ q2

??

þ2jp

x?;

j ¼ 0;1;...;

ð2:7Þ

at which Eq. (2.4) has a pair of purely imaginary roots ±ix±. Denote by

kðsÞ ¼ aðsÞ þ ixðsÞ

Y. Song, Y. Peng / Applied Mathematics and Computation 181 (2006) 1745–1757

1747

Page 4

the root of Eq. (2.4) such that

aðs?

jÞ ¼ 0;

xðs?

jÞ ¼ x?:

Substituting k(s) into (2.4) and taking the derivative with respect to s, we have

?

which, together with (2.5), leads to

?

pcosx?s?

dk

ds

??1

¼ð2k þ pÞeks

kðsk þ qÞþ

s

kðsk þ qÞ?s

k;

Redk

ds

??1

s¼s?

j

¼ Re

ð2k þ pÞeks

kðsk þ qÞ

?

?

?sx2

n

p2x2

?

s¼s?

j

þ Re

s

kðsk þ qÞ

jþ i½2x?cosx?s?

?sx2

j? 2x?sinx?s?

??

s¼s?

j

¼ Re

¼1

C

¼1

C

¼1

C

¼x2

j? 2x?sinx?s?

jþ psinx?s?

j?

?þ iqx?

j? þ qx?½2x?cosx?s?

?

þ Re

s

?sx2

j? ? s2x2

?þ iqx?

o

o

??

?½pcosx?s?

jþ psinx?s?

?

n

px?½qsinx?s?

j? sx?cosx?s?

j? þ 2x2

?¼x2

?½qcosx?s?

jþ sx?sinx?s?

j? ? s2x2

?

?þ 2x4

?? 2gx2

?? s2x2

ffiffiffiffi

?

?

?

?

C

2x2

?þ ðp2? s2? 2gÞ

¼x2

C

??

C

s2? p2þ 2g ?

D

p

þ ðp2? s2? 2gÞ

no

?

?

ffiffiffiffi

D

p

no

;

where C ¼ s2x4

sign Redk

?þ q2x2

?

?> 0. Thus, if D 5 0, we have

)

ds

ds

?

s¼sþ

j

(

¼ sign Redk

???1

s¼sþ

j

()

¼ sign

x2

þ

ffiffiffiffi

D

p

C

()

> 0

ð2:8Þ

and

sign Redk

ds

??

s¼s?

j

()

¼ sign Redk

ds

???1

s¼s?

j

()

¼ sign ?x2

?

ffiffiffiffi

D

p

C

()

< 0:

ð2:9Þ

Notice that when s = 0, Eq. (2.4) becomes

k2þ ðp þ sÞk þ g þ q ¼ 0;

two roots of which always have negative real part. In addition, note that

g2? q2¼ r2r2x2

Thus, from the previous discussions and the lemma in [10], we can obtain the following results about the

distribution of roots of Eq. (2.4).

?ða2þ a1Þða2? a1Þ;s2þ 2g ? p2¼ ?r2þ 2ra2x?r þ ðra1x?Þ2:

Lemma 2.1. Let s?

jðj ¼ 0;1;...Þ be defined as in (2.7).

(i) If at least one of the conditions (H1)–(H3) is satisfied, then all roots of Eq. (2.4) have negative real parts for

all s P 0.

(ii) If D > 0,a2> a1 and 0 < r <

a2þa1

sþ

s 2 ðsþ

(iii) If either a2< a1or a2= a1and 0 < r <

s 2 ½0;sþ

rða2þ

ffiffiffiffiffiffiffiffiffi

a2

2þa2

1

p

Þ

, then there exists a k 2 N such that when s 2 ðs?

j?1;

jÞ; j ¼ 0;1;...;k, all roots of Eq. (2.4) have negative real parts, where s?

j;s?

ffiffi

?1¼ 0, and when

jÞ; j ¼ 0;1;...;k ? 1, and s > sþ

k, Eq. (2.4) has at least one root with positive real part.

þ12

2r holds, then all roots of Eq. (2.4) have negative real parts for

p

0Þ, and Eq. (2.4) has at least one root with positive real part for s > sþ

0.

1748

Y. Song, Y. Peng / Applied Mathematics and Computation 181 (2006) 1745–1757

Page 5

Applying Lemma 2.1, we then obtain the following theorem.

Theorem 2.2. Let s?

jðj ¼ 0;1;...Þ be defined as in (2.7).

(i) At least one of the conditions (H1)–(H3) is satisfied, then the positive equilibrium E*of system (2.1) (or the

positive equilibrium x*of Eq. (1.4)) is asymptotically stable.

rða2þ

1

a2þa1

switches from stability to instability, that is, the positive equilibrium E*of system (2.1) (or the positive equi-

librium x*of Eq. (1.4)) is asymptotically stable for s 2Sk

(iii) If either a2< a1or a2= a1and 0 < r <

positive equilibrium x*of Eq. (1.4)) is asymptotically stable for s 2 ½0;sþ

(iv) If D > 0,a2> a1and 0 < r <

a2þa1

s?

a Hopf bifurcation at sþ

(ii) If D > 0,a2> a1and 0 < r <

ffiffiffiffiffiffiffiffiffi

a2

2þa2

p

Þ

, then there are k 2 N such that the stability of E*of system (2.1)

j¼0ðs?

j?1;sþ

jÞ, where s?

?1¼ 0, and unstable for

s 2Sk?1

j¼0ðsþ

j;s?

jÞ and s > sþ

k.

ffiffi

2

2r holds, then the positive equilibrium E*of system (2.1) (or the

0Þ, and unstable for s > sþ

, then system (2.1) (or Eq. (1.4)) undergoes a Hopf bifurcation at

ffiffi

p

þ1

0.

rða2þ

ffiffiffiffiffiffiffiffiffi

a2

2þa2

1

p

Þ

j. However, if either a2< a1or a2= a1and 0 < r <

j.

2

2r holds, then system (2.1) (or Eq. (1.4)) undergoes

p

þ1

3. Direction and stability of the Hopf bifurcation

In the previous section, we obtain the conditions, under which system (2.1) or Eq. (1.4) undergoes Hopf

bifurcation from the positive steady state at the critical values of s, i.e. a family of periodic solutions bifurcate

from the zero equilibrium. Using the normal form theory and center manifold reduction by Hassard et al. [9],

we are able to determine the Hopf bifurcation direction, and investigate the properties of these bifurcating

periodic solutions, for example, stability on the center manifold and period. Throughout this section, we

always assume that system (2.1) undergoes Hopf bifurcations at the critical value ~ s of s, and then ±ix is cor-

responding purely imaginary roots of the characteristic equation associated with the positive steady state E*.

Letting x1= x ? x*, x2= y ? y*, ? xiðtÞ ¼ xiðstÞ, s ¼ ~ s þ l and dropping the bars for simplification of nota-

tion, system (2.1) is transformed into an FDE in C = C([?1,0],R2) as

_ xðtÞ ¼ LlðxtÞ þ fðl;xtÞ;

where x(t) = (x1(t),x2(t))T2 R2, xt(h) = x(t + h) 2 C, and Ll:C ! R, f: R · C ! R are given, respectively, by

0

?ra2x?

r

?r

and

?

By the Riesz representation theorem, there exists a function g(h,l) of bounded variation for h 2 [?1,0], such

that

Z0

In fact, we can choose

?

where d is defined by

?

ð3:1Þ

LlðxtÞ ¼ ð~ s þ lÞ

??

x1tð0Þ

x2tð0Þ

??

þ ð~ s þ lÞ

?ra1x?

0

0

0

??

x1tð?1Þ

x2tð?1Þ

??

ð3:2Þ

fðs;xtÞ ¼ ð~ s þ lÞ

?ra1x?x1tð0Þx1tð?1Þ ? ra2x?x1tð0Þx2tð0Þ

0

?

:

ð3:3Þ

Ll/ ¼

?1

dgðh;0Þ/ðhÞ

for / 2 C:

ð3:4Þ

gðh;lÞ ¼ ð~ s þ lÞ

0

r

?ra2x?

?r

?

dðhÞ þ ð~ s þ lÞ

?ra1x?

0

0

0

??

dðh þ 1Þ;

ð3:5Þ

dðhÞ ¼

0;

1;

h 6¼ 0;

h ¼ 0:

Y. Song, Y. Peng / Applied Mathematics and Computation 181 (2006) 1745–1757

1749

Page 6

For / 2 C1([?1,0],R2), define

AðlÞ/ ¼

d/ðhÞ

dh;

R0

0;

fðl;/Þ;

h 2 ½?1;0Þ;

h ¼ 0:

?1dgðl;sÞ/ðsÞ;

(

and

RðlÞ/ ¼

h 2 ½?1;0Þ;

h ¼ 0:

?

Then system (3.1) is equivalent to

_ xt¼ AðlÞxtþ RðlÞxt;

where xt(h) = x(t + h) for h 2 [?1,0].

For w 2 C1([0,1],(R2)*), define

?dwðsÞ

R0

and a bilinear inner product

ð3:6Þ

A?wðsÞ ¼

ds;

s 2 ð0;1?;

s ¼ 0

?1dgTðt;0Þwð?tÞ;

(

hwðsÞ;/ðhÞi ¼?wð0Þ/ð0Þ ?

Z0

?1

Zh

n¼0

?wðn ? hÞ dgðhÞ/ðnÞ dn;

ð3:7Þ

where g(h) = g(h,0). Then A(0) and A*are adjoint operators. By the discussion in Section 2, we know that

?ix~ s are eigenvalues of A(0). Thus, they are also eigenvalues of A*. We first need to compute the eigenvector

of A(0) and A*corresponding to ix~ s and ?ix~ s, respectively.

Suppose that qðhÞ ¼ ða;1ÞTeihx~ sis the eigenvector of A(0) corresponding to ix~ s. Then Að0ÞqðhÞ ¼ ix~ sqðhÞ.

It follows from the definition of A(0) and (3.4) and (3.5) that

?

Thus, we can easily obtain

?

On the other hand, suppose that q?ðsÞ ¼ Dð1;bÞeisx~ sis the eigenvector of A*corresponding to ?ix~ s. By the

definition of A*and (3.4) and (3.5), we have

?

which means

q?ð0Þ ¼ Dð1;bÞ ¼ D 1;?ix þ ra1x?eix~ s

r

~ s

ix þ ra1x?e?ix~ s

?r

ra2x?

ix þ r

?

qð0Þ ¼

0

0

? ?

:

qð0Þ ¼ ða;1ÞT¼

r þ ix

r

;1

?T

:

~ s

?ix þ ra1x?eix~ s

ra2x?

?r

?ix þ r

?

q?ð0ÞðÞT¼

0

0

? ?

;

??

:

In order to assure hq*(s),q(h)i = 1, we need to determine the value of D. From (3.7), we have

hq?ðsÞ;qðhÞi ¼ Dð1;?bÞða;1ÞT?

?1

Z0

Thus, we can chose D as

Z0

Zh

n¼0

Dð1;?bÞe?iðn?hÞx~ sdgðhÞða;1ÞTeinx~ sdn

?

¼ D a þ?b ?

?1ð1;?bÞheihx~ sdgðhÞða;1ÞT

?

¼ D a þ?b þ~ sraa1x?e?ix~ s

??

D ¼

1

a þ b þ~ sraa1x?eix~ s:

1750

Y. Song, Y. Peng / Applied Mathematics and Computation 181 (2006) 1745–1757

Page 7

In the remainder of this section, we will follow the ideas and use the same notations as in [9], we first com-

pute the coordinates to describe the center manifold C0at l = 0. Let xtbe the solution of Eq. (3.1) when l = 0.

Define

zðtÞ ¼ hq?;xti;

On the center manifold C0we have

W ðt;hÞ ¼ W ðzðtÞ;? zðtÞ;hÞ;

where

W ðt;hÞ ¼ xtðhÞ ? 2RefzðtÞqðhÞg:

ð3:8Þ

W ðz;? z;hÞ ¼ W20ðhÞz2

z and ? z are local coordinates for center manifold C0in the direction of q*and ? q?. Note that W is real if xtis

real. We consider only real solutions. For the solution xt2 C0of (3.1), since l = 0, we have

_ zðtÞ ¼ ix~ sz þ hq?ðhÞ;Fð0;W ðz;? z;hÞ þ 2RefzqðhÞgÞi

¼ ix~ sz þ ? q?ð0Þfð0;W ðz;? z;0Þ þ 2Refzqð0ÞgÞ ¼

We rewrite this equation as

2þ W11ðhÞz? z þ W02ðhÞ? z2

2þ ???;

defix~ sz þ ? q?ð0Þf0ðz;? zÞ:

_ zðtÞ ¼ ix~ szðtÞ þ gðz;? zÞ

with

gðz;? zÞ ¼ ? q?ð0Þf0ðz;? zÞ ¼ g20

Noticing xtðhÞ ¼ ðx1tðhÞ;x2tðhÞ;x3tðhÞÞ ¼ W ðt;hÞ þ zqðhÞ þ zqðhÞ and qðhÞ ¼ ða;1ÞTeihx~ s, we have

x1tð0Þ ¼ az þ az þ Wð1Þ

20ð?1Þz2

20ð0Þz2

z2

2þ g11z? z þ g02

? z2

2þ g21

z2? z

2þ ???ð3:9Þ

20ð0Þz2

2þ Wð1Þ

11ð0Þz? z þ Wð1Þ

02ð0Þ? z2

2þ Oðjðz;? zÞj3Þ;

02ð?1Þ? z2

x1tð?1Þ ¼ ae?ix~ sz þ aeix~ s? z þ Wð1Þ

2þ Wð1Þ

11ð?1Þz? z þ Wð1Þ

02ð0Þ? z2

2þ Oðjðz;? zÞj3Þ;

x2tð0Þ ¼ z þ? z þ Wð2Þ

2þ Wð2Þ

11ð0Þz? z þ Wð2Þ

2þ Oðjðz;? zÞj3Þ:

Thus, from (3.9), we have

gðz;? zÞ ¼ ? q?ð0Þf0ðz;? zÞ ¼ ~ sDð1;?bÞ

?ra1x?x1tð0Þx1tð?1Þ ? ra2x?x1tð0Þx2tð0Þ

0

??

¼ ?~ sDrx? a1 az þ ? az þ Wð1Þ

?

þ a2 az þ ? az þ Wð1Þ

?

¼ ?~ sDrx? 2 a1a2e?ix~ sþ a2a

?

þ a2 2aWð2Þ

20ð0Þz2

20ð?1Þz2

2þ Wð1Þ

11ð0Þz? z þ Wð1Þ

02ð0Þ? z2

02ð?1Þ? z2

2þ Oðjðz;? zÞj3Þ

???

?

ae?ix~ sz þ ? aeix~ s? z þ Wð1Þ

?

z þ? z þ Wð2Þ

?

þ a1 2aWð1Þ

?

2þ Wð1Þ

11ð?1Þz? z þ Wð1Þ

02ð0Þ? z2

02ð0Þ? z2

2þ Oðjðz;? zÞj3Þ

?

??

i

20ð0Þ

?

?

20ð0Þz2

2þ Wð1Þ

11ð0Þz? z þ Wð1Þ

2þ Oðjðz;? zÞj3Þ

?

20ð0Þz2

2þ Wð2Þ

11ð0Þz? z þ Wð2Þ

?z2

20ð?1Þ þ 2ae?ix~ sWð1Þ

2þ Oðjðz;? zÞj3Þ

2þ 2 a1jaj2Refeix~ sg þ a2Refag

h

z? z þ 2 a1? a2eix~ sþ a2? a

?

.

??? z2

2

?

11ð?1Þ þ ? aWð1Þ

11ð0Þ þ ? aeix~ sWð1Þ

?iz2? z

h

11ð0Þ þ ? aWð1Þ

20ð0Þ þ 2Wð1Þ

11ð0Þ þ Wð2Þ

20ð0Þ

2þ ???

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Comparing the coefficients with (3.9), we get

?

g11¼ ?2~ sDrx? a1jaj2Refeix~ sg þ a2Refag

g02¼ ?2~ sDrx? a1a2eix~ sþ a2a

g21¼ ?2~ sDrx? a1 2aWð1Þ

þa2 2aWð2Þ

Since there are W20(h) and W11(h) in g21, in the sequel, we shall compute them.

From (3.6) and (3.8), we have

?

where

g20¼ ?2~ sDrx? a1a2e?ix~ sþ a2a

?

n

?;

?

;

??;

11ð?1Þ þ aWð1Þ

11ð0Þ þ aWð1Þ

20ð?1Þ þ 2ae?ix~ sWð1Þ

20ð0Þ þ 2Wð1Þ

11ð0Þ þ aeix~ sWð1Þ

?o

20ð0Þ

??

11ð0Þ þ Wð2Þ

20ð0Þ

?

:

ð3:10Þ

_W ¼ _ xt? _ zq ?_? z? q ¼

AW ? 2Ref? q?ð0Þf0qðhÞg;

AW ? 2Ref? q?ð0Þf0qð0Þg þ f0;

h 2 ½?1;0Þ;

h ¼ 0;

¼

defAW þ Hðz;? z;hÞ;

ð3:11Þ

Hðz;? z;hÞ ¼ H20ðhÞz2

Expanding the above series and comparing the corresponding coefficients, we obtain

2þ H11ðhÞz? z þ H02ðhÞ? z2

2þ ???:

ð3:12Þ

ðA ? 2x~ sÞW20ðhÞ ¼ ?H20ðhÞ;AW11ðhÞ ¼ ?H11ðhÞ;...:

From (3.11), we know that for h 2 [?1,0),

Hðz;? z;hÞ ¼ ?? q?ð0Þf0qðhÞ ? q?ð0Þf0? qðhÞ ¼ ?gqðhÞ ? ? g? qðhÞ:

Comparing the coefficients with (3.12) gives that

ð3:13Þ

ð3:14Þ

H20ðhÞ ¼ ?g20qðhÞ ? ? g02? qðhÞ;

ð3:15Þ

and

H11ðhÞ ¼ ?g11qðhÞ ? ? g11? qðhÞ:

From (3.13), (3.15) and the definition of A , it follows that

ð3:16Þ

_W20ðhÞ ¼ 2ix~ sW20ðhÞ þ g20qðhÞ þ ? g02? qðhÞ:

Notice that qðhÞ ¼ ð1;a;bÞTeihxksk, hence

W20ðhÞ ¼ig20

x~ sqð0Þeihx~ sþi? g02

3x~ s? qð0Þe?ihx~ sþ E1e2ihx~ s;

ð3:17Þ

where E1¼ ðEð1Þ

Similarly, from (3.13) and (3.16), we can obtain

W11ðhÞ ¼ ?ig11

where E2¼ ðEð1Þ

In what follows, we shall seek appropriate E1and E2. From the definition of A and (3.13), we obtain

Z0

and

Z0

1;Eð2Þ

1Þ 2 R2is a constant vector.

x~ sqð0Þeihx~ sþi? g11

x~ s? qð0Þe?ihx~ sþ E2;

ð3:18Þ

2;Eð2Þ

2Þ 2 R2is also a constant vector.

?1

dgðhÞW20ðhÞ ¼ 2ix~ sW20ð0Þ ? H20ð0Þð3:19Þ

?1

dgðhÞW11ðhÞ ¼ ?H11ð0Þ;

ð3:20Þ

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where g(h) = g(0,h). From (3.11) and (3.12), we have

H20ð0Þ ¼ ?g20qð0Þ ? ? g02? qð0Þ ? 2~ srx?

a1a2e?ix~ sþ a2a

0

??

ð3:21Þ

and

H11ð0Þ ¼ ?g11qð0Þ ? ? g11? qð0Þ ? 2~ srx?

a1jaj2Refeix~ sg þ a2Refag

0

!

:

ð3:22Þ

Substituting (3.17) and (3.21) into (3.19) and noticing that

Z0

and

Z0

we obtain

Z0

which leads to

?

ix~ sI ?

?1

eihx~ sdgðhÞ

??

qð0Þ ¼ 0

?ix~ sI ?

?1

e?ihx~ sdgðhÞ

??

? qð0Þ ¼ 0;

2ix~ sI ?

?1

e2ihx~ sdgðhÞ

??

E1¼ ?2rx?

a1a2eix~ sþ a2a

0

??

;

2ix þ ra1x?e?2ix~ s

?r

It follows that

ra2x?

2ix þ r

?

E1¼ ?2rx?

a1a2eix~ sþ a2a

0

??

:

Eð1Þ

1 ¼?2rx?ðr þ 2ixÞða1a2eix~ sþ a2aÞ

A

and

Eð2Þ

1 ¼?2rx?rða1a2eix~ sþ a2aÞ

A

;

where

A ¼

2ix þ ra1x?e?2ix~ s

?r

ra2x?

2ix þ r

????????:

Similarly, substituting (3.18) and (3.22) into (3.20), we can obtain

ra1x?

?r

ra2x?

r

??

E2¼ 2rx?

a1jaj2Refeix~ sg þ a2Refag

0

!

:

and hence,

Eð1Þ

2 ¼ Eð2Þ

2 ¼

2 a1jaj2Refeix~ sg þ a2Refag

a1þ a2

??

:

Therefore, we can determine W20(0) and W11(0) from (3.17) and (3.18). Furthermore, we can determine g21.

Therefore, each gijin (3.9) is determined by the parameters and delay in (3.10). Thus, we can compute the fol-

lowing values:

Y. Song, Y. Peng / Applied Mathematics and Computation 181 (2006) 1745–1757

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c1ð0Þ ¼

i

2~ sx

g11g20? 2jg11j2?jg02j2

3

!

þg21

2;

l2¼ ?Refc1ð0Þg

Refk0ð~ sÞg;

b2¼ 2Refc1ð0Þg;

T2¼ ?Imfc1ð0Þg þ l2Imfk0ð~ sÞg

~ sx

;

ð3:23Þ

which determine the quantities of bifurcating periodic solutions in the center manifold at the critical value ~ s,

i.e., l2determines the directions of the Hopf bifurcation: if l2> 0(l2< 0), then the Hopf bifurcation is super-

critical (subcritical) and the bifurcating periodic solutions exist for s > ~ sðs < ~ sÞ; b2determines the stability of

the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if b2< 0 (b2> 0); and

T2determines the period of the bifurcating periodic solutions: the period increase (decrease) if T2> 0 (T2< 0).

These, together with (2.8), (2.9), Lemma 2.1 and Theorem 2.2, imply the following.

Theorem 3.1. (i) Suppose that D > 0,a2> a1and 0 < r <

s ¼ sþ

inverse conclusion holds for s ¼ s?

supercritical) if Re{c1(0)} < 0 (respectively, Re{c1(0)} > 0). On the center manifold, the Hopf bifurcating periodic

solution at E*for s ¼ s?

holds, then system (2.1) undergoes a Hopf bifurcation at sþ

(respectively, subcritical) if Re{c1(0)} < 0 (respectively, Re{c1(0)} > 0). On the center manifold, the Hopf

bifurcating periodic solution are stable(unstable) if b2< 0 (b2> 0).

rða2þ

ffiffiffiffiffiffiffiffiffi

a2

2þa2

1

p

a2þa1

Þ

. Then the Hopf bifurcations of (2.1) at

jare supercritical (respectively, subcritical) if Re{c1(0)} < 0 (respectively, Re{c1(0)} > 0). However, the

j, namely, the Hopf bifurcations of (2.1) at s ¼ s?

jare subcritical (respectively,

jare stable(unstable) if b2< 0 (b2> 0). (ii) If either a2< a1or a2= a1and 0 < r <

j, and the Hopf bifurcations are supercritical

ffiffi

2

2r

p

þ1

A numerical example. From the above algorithm, we know that if the values of r,r,ai(i = 1,2) and s are

given, then we can compute the values of l2and b2determine the stability and direction of periodic solutions

bifurcating from the positive equilibrium at the critical point s?

results by considering the system:

j. In the sequel, we illustrate the validity of the

_ xðtÞ ¼ xðtÞ½1 ? 2xðt ? sÞ ? yðtÞ?;

_ yðtÞ ¼ xðtÞ ? yðtÞ;

which has a positive equilibrium E?¼ ð1

2jp

0:797728, the positive equilibrium E*is stable when s < sþ

bifurcation at sþ

?

ð3:24Þ

3;1

3Þ. It follows from Section 2 and Theorem 2.2 that sþ

0(see Figs. 1–3), and system (3.24) undergoes a Hopf

j. Further, from the above process, we can determine the stability and direction of periodic

j¼:2:77303þ

Fig. 1. Behavior of x in system (3.24) with s = 2.3 and the initial value x0= 0.8.

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Y. Song, Y. Peng / Applied Mathematics and Computation 181 (2006) 1745–1757

Page 11

solutions bifurcating from the positive equilibrium at the critical point sþ

we can compute c1(0) G ?0.399728 ? 0.00886001I. It then follows that l2> 0 and b2< 0. Therefore, the

bifurcating periodic solutions exists at least for the value s slightly large than sþ

odic orbits are orbitally asymptotically stable, as depicted in Figs. 4–6.

j. For instance, when s ¼ sþ

0¼:2:77303,

0and the corresponding peri-

Fig. 2. Behavior of y in system (3.24) with s = 2.3 and the initial value y0= 0.8.

Fig. 3. Phase portrait of system (3.24) with s = 2.3, (x0,y0) = (0.8,0.8). The positive equilibrium (x*,y*) of system (3.24) is stable for

s < sþ

0.

Fig. 4. Periodic oscillation of x in system (3.24) with s = 2.8 and the initial value x0= 0.8.

Y. Song, Y. Peng / Applied Mathematics and Computation 181 (2006) 1745–1757

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4. Conclusions

The Logistic equation with one or two discrete delays has been widely studied by authors. For the delayed

Logistic equation (1.1), it is well known that (i) the positive equilibrium x ? k is asymptotically stable for any

s 2 ½0;p

x ? k at s ¼1

the distributed delay (i.e., a1> a2), then Eq. (1.4) almost has the same dynamics as the Logistic equation (1.1)

with a discrete delay. However, Theorem 2.2 shows that the dynamics of the Logistic equation with discrete

and distributed delays are more rich and interesting. For example, the phenomena of stability switches occur if

D > 0,a2> a1and 0 < r <

a2þa1

2rÞ, and unstable for s >p

rfp

2r, and (ii) Eq. (1.1) undergoes a Hopf bifurcation from the positive equilibrium

2þ 2jpg. From Theorem 2.2, we obtain that if the effect of the discrete delay is large than that of

rða2þ

ffiffiffiffiffiffiffiffiffi

a2

2þa2

1

p

Þ

.

References

[1] G.S. Jones, The existence of periodic solution of f

[2] S.N. Chow, J. Mallet-Paret, Integral averaging and bifurcation, J. Differential Equations 26 (1977) 112–159.

[3] A. Stech, Hopf bifurcation calculations for functional differential equations, J. Math. Anal. Appl. 109 (1985) 472–491.

[4] I. Gyo ¨ri, G. Ladas, Oscillation theory of delay differential equations. With applications, Oxford Mathematical Monographs, Oxford

Science Publications. The Clarendon Press, Oxford University Press, New York, 1991.

[5] J. Hale, S.M. Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.

[6] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.

0(x) = ?af(x ? 1)[1 + f(x)], J. Math. Anal. Appl. 5 (1962) 435–450.

Fig. 6. Phase portrait of system (3.24) with s = 2.8. Here initial values are taken as (0.4,0.4) and (0.8,0.8), respectively. Clearly, this

numerical simulation shows that the bifurcating periodic solution is asymptotically stable.

Fig. 5. Periodic oscillation of y in system (3.24) with s = 2.8 and the initial value y0= 0.8.

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Page 13

[7] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and its Applications,

vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1992.

[8] J.M. Cushing, Integro-differential Equations and Delay Models in Population Dynamics, Springer, Heidelberg, 1977.

[9] B.D. Hassard, N.D. Kazarinoff, Y.H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge,

1981.

[10] K. Cooke, Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl. 86 (1982) 592–627.

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